Number of $32$-character alphanumeric strings with certain conditions

Question

I'm seeking a solution of one of the most complicated Math problem of my life.

Here it is :

First we need to figure out how many strings of set [a-zA-Z0-9] (Which is 26 Small Letters, 26 Capital Letters, and 0 to 9 digits] Are possible to construct of length 32 characters.

Then we need to subtract these 3 out of our result.

1. Possible 32 character strings which has only small letters. [a-z]
2. Possible 32 character strings which has only big letters. [A-Z]
3. Possible 32 character strings which has only digits. [0-9]

Let me know if any questions.

Answer 1

Please let me know if I have grossly misinterpreted the problem.

It seems clear to me that we are allowed to repeat letters and numbers. Then, for each slot in our $32$-character string, we have $62$ choices, adding all of the options together ($26$ letters, both upper and lower case, and the $10$ numbers).

Since there are $32$ characters and repetition is allowed, we come to $62^{32} = 2272657884496751345355241563627544170162852933518655225856$ possibilities. Good lord. According to Wolfram Alpha, that is roughly 43 million times the number of possible chess positions. Good lord.

For strings made entirely of capital letters, we have $26^{32}=1901722457268488241418827816020396748021170176$ possibilities. This is also true for the lowercase letters.

Similarly, there are $10^{32}=100000000000000000000000000000000$ possible $32$-element strings of just digits.

So, your answer, apparently, is $62^{32}-2(26)^{32}-10^{32} = 2272657884492947900440704487144706514530812140022612885504$.

Answer 2

Let $U,L,D$ be the sizes of the sets of strings containing only uppercase letters (26 usable characters), lowercase letters (26) or digits (10) respectively. Then the number of admissible strings is $$62^{32}-U-L-D=62^{32}-2\cdot26^{32}-10^{32}$$ $$=2.272\dots×10^{57}$$

Answer 3

First, there are the following possibilities for strings:

• 32-character strings containing only small letters
• 32-character strings containing only capital letters
• 32-character strings containing only digits
• 32-character strings containing only small letters and capital letters
• 32-character strings containing only small letters and digits
• 32-character strings containing only capital letters and digits
• 32-character strings containing small letters, capital letters and digits

So, by removing the first 3 categories, what you'll end up with is the total number of 32-character strings which combine 2 or more of the categories. Make sure that's the answer you want.

26 small letters + 26 capital letters + 10 digits = 62 characters total. For the first character in a string, you obviously have 62 possibilities. Since the next character can be the same or different, you have 62 possibilities there as well, or $62^{2}$ possibilities for 2 characters. Adding a 3rd character gives $62^{3}$ possibilities. For 32 characters, then, there's $62^{32}$ possibilities.

There are 26 possible small letters, so by the same logic, you'll get $26^{32}$ possibilities for 32 character strings containing all small letters.

The same logic and math applies to capital letter-only strings, so there's another $26^{32}$ possibilities.

For 10 different digits, 0 through 9, this works out to $10^{32}$ possibilities.

The equation you want, therefore, is:

$$62^{32}-26^{32}-26^{32}-10^{32}$$

You can get the final total on, say, Wolfram|Alpha.

Again, what you're getting is the number of possible 32-character strings containing combinations of 2 or more of the "small letter", "capital letter" and "digit" categories. It's not only important to get the total, but to understand the meaning of the result, as well.